Calculating the Travel Time of High-Energy Particles Across Our Galaxy

[senatorarmstrong]

Homework Statement

Our galaxy is about 10^5 light-years across, and the most energectic particles known have an energy of about 10^19 eV. How long would it take a proton with this energy to traverse the galaxy as measured from (a) the galaxy and (b) the particle?

Homework Equations

I attempted to use the relationship between energy and momentum. E² = p²*c² + m²*c⁴

I also tried solving the problem with relativistic kinetic energy. K = mc²(γ-1)

The Attempt at a Solution

I used both equations and got similar results...

I figured what I could do is use the energy given and then solve for u in γ. Knowing the relativistic velocity, I could then find how long it would take this proton to zip across the milky-way.

If I am solving for u in γ, I am obviously solving for the velocity of the particle (b) and not the velocity of the particle with respect the galaxy (a).

What throws me off is using the two equations up there, I got u to be equal to c. When I used the kinetic energy equation, u > c! Can't be...

Any hints?

Thanks!

 

[kuruman]

Please show exactly what you did. Then we will be able to guide your thinking.